Me and my coauthor are writing a (math) paper with a third person who initially has been very active, mainly in the form of conversations and planning, and then after writing some parts, disappearead and proceeded to ignore every attempt to get in touch with them.
Let me note we are aware they've been in contact with other people we know (and active on social media). So, as far as we understand, they simply chose to ignore us. Everything was running smoothly and we didn't have a fight or a discussion so we are very confused (and annoyed) by this behaviour. But anyway, we carried on writing our paper.
Now we are very close to submission (on arXiv, for the moment) and we are pondering what to make of the third author's name. We thought of removing their name from the authors list since their behaviour has been quite unacceptable. We ended up doing the great share of the work ourselves, and there hasn't been any reasonable motivation from their part on why this is so. On the other hand, they provided valuable input and even wrote up some parts which, even after some revisions, will end up in the paper, making up, say, 5-10% of the content. These are things they wrote before disappearing. So it seems correct to attribute authorship where due.
On a more pedantic note, they don't know the state of the paper we are going to submit so, as far as we know, we might not have their permission to put their name on it. We are going to ask them but, at this point, it's unlikely they are going to reply.
Hence, what shall we do? Is there a middle ground like mentioning their contribution in the 'acknowledgments', perhaps indicating they have been closely involved in the writing of some parts?
EDIT: after some pondering, we changed the paper to remove the small written contributions (which were proofs) made by the ghosting coauthor and replaced with alternative proofs (using a completely different approach) written by us or referenced from literature. We plan to still acknowledge them in our paper, for the parts they've been involved to, but not to have it in the coauthors list anymore.